# Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and is drected towards it.

• The motion is periodic and repetitive.

• The acceleration is directly proportional to the displacement and is always directed towards the equilibrium position.

• The velocity is maximum at the equilibrium position and minimum at the extreme positions.

• The displacement, velocity, and acceleration are all sinusoidal functions of time.

• The period of oscillation is independent of the amplitude.

Examples of SHM include: A mass attached to a spring and oscillating vertically or a A pendulum swinging back and forth.

``Displacement: x = A cos(ωt + φ)Velocity: v = -Aω sin(ωt + φ)Acceleration: a = -Aω^2 cos(ωt + φ)``

where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle.

## Energy in SHM

The total mechanical energy of a system undergoing SHM is constant and is the sum of kinetic and potential energy.

• Total energy: E = 1/2 kA^2

• Kinetic energy: K = 1/2 mv^2

• Potential energy: U = 1/2 kx^2

where k is the spring constant, m is the mass, v is the velocity, and x is the displacement.

# Force and Amplitude

## Force in Simple Harmonic Motion

Since the block is acceleration and deceleration, there must be some force that is making it do so. This this cause, the spring exerts a force on the block.

• It is a vector quantity, meaning it has both magnitude and direction.

• The SI unit of force is Newton (N).

• Force can cause an object to accelerate, change direction, or deform.

``F = -kx``

Also known as Hooke’s Law, the k is the called the spring constant and tells us how strong the spring is. The greater the K, the stiffer the spring actually is.

## Amplitude

• Amplitude is a measure of the magnitude of a wave.

• It is the maximum displacement of a particle from its equilibrium position in a wave.

• Amplitude is measured in meters (m) for a mechanical wave and in volts (V) for an electromagnetic wave.

• The amplitude of a wave determines its intensity and energy.

• The amplitude of a wave can also affedct th eforc e.

# Period and Frequency

## Period

• The period of a wave is the time it takes for one complete cycle of the wave to occur.

• It is denoted by the symbol T and is measured in seconds (s).

• The period is inversely proportional to the frequency of the wave.

• Mathematically, T = 1/f, where f is the frequency of the wave.

## Frequency

• The frequency of a wave is the number of complete cycles of the wave that occur in one second.

• It is denoted by the symbol f and is measured in Hertz (Hz).

• The frequency is directly proportional to the energy of the wave.

• Mathematically, f = 1/T, where T is the period of the wave.

# Pendulums

In a system, if the spring is used the period will increase with objects mass and decrease with as a greater spring constant increases. This is because a mass will resist acceleration and a large spring constant will make the spring exert more force.

• Object's mass affects the period of the spring, not the pendulum

• Mass can act as either inertial or gravitational

• Gravitational mass accelerates due to the force of gravity between it and the Earth

• All objects fall at the same rate due to gravitational mass

• Inertial mass is not pushed by gravity but by external forces like the spring force

• Inertial mass can accelerate at different rates depending on the force applied

Pendulums are a common topic in AP Physics Unit 7, which covers simple harmonic motion. A pendulum is a weight suspended from a pivot point that swings back and forth due to gravity. The period of a pendulum (the time it takes to complete one full swing) is determined by the length of the pendulum and the acceleration due to gravity. The equation for the period of a pendulum is T=2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. Pendulums are used in many applications, such as clocks and seismometers.